Near-Popular Matchings in the Roommates Problem

Abstract. Our input is a graph G = (V,E) where each vertex ranks its neighbors in a strict order of preference. The problem is to compute a matching in G that captures the preferences of the vertices in a popular way. Matching M is more popular than matching M ′ if the number of vertices that prefer M to M ′ is more than those that prefer M ′ to M. The unpopularity factor of M measures by what factor any matching can be more popular than M. We show that G always admits a matching whose unpopularity factor is O(log |V |) and such a matching can be computed in linear time. In our problem the optimal matching would be a least unpopularity factor matching- we show that computing such a matching is NP-hard. In fact, for any > 0, it is NP-hard to compute a matching whose unpopularity factor is at most 4/3 − of the optimal